Asymptotic stability of nonlinear functional differential equations

Abstract

In this paper, we study asymptotic stability of solutions of the following functional differential equation: x'(t) = a(t)x(t) + b(t)x(p(t)) + f ( t , x(t), x(p(t))) x(O) = xo. (1) Here a,b : [0, oo) ~ C, f : [0, oo) × C × C ~ C are continuous functions, x0 E C and p : [0, oo) ~ [0, oo) is a continuously differentiable function such that the following conditions hold: (i) limt_oo p(t) = oo (ii) p(t) < t, t E [0, co) (iii) 0 * limt_oo p' (t) < oo. The technical condition (iii) holds, for example, for the case of proportional delays. Our results can be generalised for equations with finite number of delays and also for systems. To achieve the latter, use of logarithmic norm (cf. [1]) of a matrix is essential. For the ease of presentation, however, we consider only the simplest case. Earlier, asymptotic stability of solutions of the neutral functional differential equation.